NORMAL, CONSTANT VARIANCE; proc genmod data=a

*IN THIS CODE, X = EXPOSURE, {C1-C3} ARE CONFOUNDERS;

*NORMAL, CONSTANT VARIANCE;

proc genmod data=a;

model x = c1 c2 c3 / maxiter=100;

*variance of x is sse/df = valuedf;

ods output Modelfit=ss(where=(criterion="Deviance"));

output out=x xbeta=xb;*predicted mean is xb;

data ss;set ss;merg=1;

data x;set x;merg=1;

data a1;

merge x ss;

by merg;

sd_x=sqrt(valuedf);*standard deviation;

pdf_den = pdf("normal",x,xb,sd_x);*denominator density value;

drop criterion df value valuedf;

run;

*t, CONSTANT VARIANCE;

proc genmod data=a;

model x = c1 c2 c3 / maxiter=100;

output out=x stdresdev=e;

data a1;

set x;

pdf_den = pdf("t",e,1);*denominator density value with 1 df;

run;

*NORMAL, HETEROSCEDASTIC VARIANCE, TRUNCATED NORMAL, HETEROSCEDASTIC VARIANCE;

*QLIM CODE PROVIDED BY DR MIGUEL HERNAN;

proc qlim data=a;

*TO FIT TRUNCATED NORMAL, UNCOMMENT BELOW;

model x = c1 c2 c3 ;* / truncated(lb=5 ub=30) ; hetero x ~ c1 c2 c3 / link = exp noconst;

nloptions tech=newrap maxiter=1000 maxfunc = 2000;

output out=x predicted errstd;

data a1;

set x;

pdf_den = pdf("normal",x,p_x,errstd_x);

*above gives normal heteroscedastic;

p_n = pdf("normal",x,p_x,errstd_x) ; p_lb = cdf("normal",5,p_x,errstd_x);

p_ub = cdf("normal",30,p_x,errstd_x);

pdf_den_trunc = p_n / (p_lb - p_ub);

*above gives truncated normal heteroscedastic;

run;

*GAMMA;

proc genmod data=a;

model x = c1 c2 c3 / dist=gamma link=log maxiter=100;

ods output ParameterEstimates=s(where=(Parameter="Scale"));

output out=x p=xb;

data s;set s;merg=1;rename estimate=scale;

drop parameter df stderr lowerwaldcl upperwaldcl chisq probchisq;

data x; set x;merg=1;

data a1;

merge x s;

by merg;

*determine Gamma shape parameter;

lambda = xb/scale;

*estimate denominator density;

pdf_den = pdf("gamma",x,scale,lambda);

run;

*to obtain numerator density values for all of the above, fit models without c1, c2, or c3 and repeat data step a1;

*QUANTILE BINNING;

proc rank data=a out=x ties=low groups=10;

var x;ranks x_r;

proc logistic data=x desc;

model x_r = c1 c2 c3 / maxiter=100;

output out=a1 predprobs=i;

run;quit;run;

data a1;

set a1;

array ii(*) ip_1-ip_10;

*ip_1-ip_10 are predicted probabilities of being in category 1 to 10;

do j=1 to 10;

*set density to predicted probability of observed exposure category;

if x_r = j then pdf_den=ii(j);

end;

drop j ip_1-ip_10;

run;quit;run;

*set numerator density value to 1/j;

*END;

 

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(A)

QB20 QB15 QB10 G(10,90) G(5,95) G(1,99) G t5 t3 t1 NHu(10,90) NHu(5,95) NHu(1,99) NHu NHh(10,90) NHh(5,95) NHh(1,99) NHh NHc(10,90) NHc(5,95) NHc(1,99) NHc N(10,90) N(5,95) N(1,99) N

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(B)

−0.5 0 0.25 0.5

QB20 QB15 QB10 G(10,90) G(5,95) G(1,99) G t5 t3 t1 NHu(10,90) NHu(5,95) NHu(1,99) NHu NHh(10,90) NHh(5,95) NHh(1,99) NHh NHc(10,90) NHc(5,95) NHc(1,99) NHc N(10,90) N(5,95) N(1,99) N

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******************************************************************************;

"CONSTRUCTING INVERSE PROBABILITY WEIGHTS FOR A CONTINUOUS EXPOUSRE"

**AUTHOR: ASHLEY ISAAC NAIMI (ashley.naimi@mcgill.ca)

**VERSION: 3.3

**PURPOSE: GENERATE SIMULATED DATA AND ASSESS DIFFERENT METHODS TO CONSTRUCT IPW

******************************************************************************;

%inc "c:\ain\header.txt";

%inc "c:\ain\spline.txt";

proc printto log="Y:\Documents\Research\Papers\ConstructingWeights\20Jun13\ipw.sim2.log";run;

proc printto

print="Y:\Documents\Research\Papers\ConstructingWeights\20Jun13\ipw.sim2.lst";run;

*IMPORT QUEBEC BIRTH FILE DATA TO GENERATE SIMULATED DATA. KEEP ONLY MATERNAL AND PATERNAL AGE AND PARITY;

proc import datafile="Y:\Documents\Research\Papers\ConstructingWeights\20Jun13\ipimoms.txt"

out=moms dbms=dlm replace;

delimiter='09'x;

run;

data moms;set moms;id = _N_;keep mage page parity id;run;

*SELECT 3000 IID RANDOM SAMPLES OF QUEBEC BIRTH FILE DATA TO GENERATE SIMULATED EXPOSURE AND OUTCOME DATA;

proc surveyselect data=moms

out=a noprint method=srs reps=50 seed=675847 sampsize=1500;

run;

proc print data=a (obs=5);run;

*CREATE DISJOINT INDICATOR VARIALBES FOR PARITY;

data a;

set a;

array par(*) parity1-parity5;

do pp = 1 to 5;

par(pp) = (parity=pp);

end;

run;

*GENERATE TWO SIMULATED EXPOSURES (x2a and x2c) AND OUTCOMES AS A FUNCTION OF QUEBEC BIRTH FILE DATA COVARIATES;

data a;

set a;

call streaminit(789654);

r = rand("bernoulli",.5);

mu = .025*mage + .0025*page + .00125*mage*page

- .21*parity2 - .22*parity3 - .45*parity4 - .45*parity5;

x2a = rand("normal",15 + mu, 1.5);*r + exp(.95*rannor(987))*(1-r);*exp(mu + rand("normal",0,.25));*CONTAMINATED-NORMAL DISTRIBUTION: HOMOSCEDASTIC;

if x2a < 0 then x2a = 0;

jitter = rand("normal");

x2c = (rand("Poisson",mu) + jitter);*POISSON DISTRIBUTION:

HETEROSCEDASTIC WITH MEAN = VARIANCE;

if x2c < 0 then x2c = 0;

p2a = 1/(1+exp(-(-11.5 + log(1.25)*x2a + log(1.7)*sqrt(mage) + log(1.5)*sqrt(page)

+

log(0.75)*parity2 + log(0.8)*parity3 + log(0.85)*parity4 + log(0.9)*parity5)));

y2a = rand("bernoulli",p2a);

p2c = 1/(1+exp(-(-8.05 + log(1.25)*x2c + log(1.7)*sqrt(mage) + log(1.5)*sqrt(page)

+

log(0.75)*parity2 + log(0.8)*parity3 + log(0.85)*parity4 + log(0.9)*parity5)));

y2c = rand("bernoulli",p2c);

smage = sqrt(mage);

spage = sqrt(page);

logx2a = x2a;

llx2a = log(x2a);

logx2c = log(x2c+.001);

x2cind = (x2c>0);

pind = sum(of parity2-parity5);

cens=1;

magepage = mage*page;

rename replicate=h ; label replicate = ' ' ; run;

proc means data=a min max mean median q1 q3 var nmiss maxdec=3;

var x2a x2c logx2c y2a y2c mage page parity2-parity5;run;

*DETERMINE MARGINAL TRUE VALUE USING METHOD OF MARGINAL STANDARDIZATION: Localio AR, ET AL.

Journal of Clinical Epidemiology. 2007;

data check;

set a;

linear2anum = -11.5 + log(1.25)*17.984 + log(1.7)*sqrt(mage) + log(1.5)*sqrt(page) + log(0.75)*parity2 + log(0.8)*parity3 + log(0.85)*parity4 + log(0.9)*parity5;

p2anum = 1/(1+exp(-(linear2anum)));

linear2aden = -11.5 + log(1.25)*16.984 + log(1.7)*sqrt(mage) + log(1.5)*sqrt(page) + log(0.75)*parity2 + log(0.8)*parity3 + log(0.85)*parity4 + log(0.9)*parity5;

p2aden = 1/(1+exp(-(linear2aden)));

linear2cnum = -8.05 + log(1.25)*2.067 + log(1.7)*sqrt(mage) + log(1.5)*sqrt(page) + log(0.75)*parity2 + log(0.8)*parity3 + log(0.85)*parity4 + log(0.9)*parity5;

p2cnum = 1/(1+exp(-(linear2cnum)));

linear2cden = -8.05 + log(1.25)*1.067 + log(1.7)*sqrt(mage) + log(1.5)*sqrt(page) + log(0.75)*parity2 + log(0.8)*parity3 + log(0.85)*parity4 + log(0.9)*parity5;

p2cden = 1/(1+exp(-(linear2cden)));

run;

proc means data=check noprint mean;

var p2anum p2aden;

var p2cnum p2cden;

output out=meancheck mean=p2anum p2aden p2cnum p2cden;

data meancheck;set meancheck;

truex2a = log((p2anum/(1-p2anum))/(p2aden/(1-p2aden)));truex2c = log((p2cnum/(1- p2cnum))/(p2cden/(1-p2cden)));

merg = 1;

keep merg truex2a truex2c;

run;

*CONSTRUCTING IPWS;

*MODEL MARGINAL AND CONDITIONAL MEAN OF EXPOSURE TO CONSTRUCT IPW;

*FOR X2C MODEL ZERO-INFLATED PORTION;

ods select none;

proc logistic data=a;

by h;

title "Continuous Exposure Model: Numerator Zero Portion";

model x2cind = / maxiter=100;

output out=nx2cbin p=nx2cbin;

proc logistic data=a;

by h;

title "Continuous Exposure Model: Denominator Binary Portion";

model x2cind = mage page mage*page parity2-parity5 / maxiter=100;

output out=dx2cbin p=dx2cbin;

*NORMAL MODEL;

proc genmod data=a;

by h;

title "Continuous Exposure Model: Numerator";

model logx2a = / maxiter=100;

ods output Modelfit=ssnx2a(where=(criterion="Deviance"));

output out=nx2aa xbeta=nxba stdresdev=nx2a;run;

data nx2a;

merge nx2aa ssnx2a;

by h;

varnx2a=sqrt(valuedf);

drop criterion df value valuedf;

run;

proc genmod data=a;

by h;*where x2c>0;

title "Continuous Exposure Model: Numerator";

model logx2c = / maxiter=100;

ods output Modelfit=ssnx2c(where=(criterion="Deviance"));

output out=nx2cc xbeta=nxbc stdreschi=nx2c;

data nx2c;

merge nx2cc ssnx2c;

by h;

varnx2c=sqrt(valuedf);

drop criterion df value valuedf;

run;

proc genmod data=a;

by h;

title "Continuous Exposure Model: Denominator";

model x2a = mage page mage*page parity2-parity5 / maxiter=100;

ods output Modelfit=ssdx2a(where=(criterion="Deviance"));

output out=dx2aa xbeta=dxba stdresdev=dx2a;

data dx2a;

merge dx2aa ssdx2a;

by h;

vardx2a=sqrt(valuedf);

drop criterion df value valuedf;

run;

proc genmod data=a;

by h;*where x2c>0;

title "Continuous Exposure Model: Denominator";

model logx2c = mage page mage*page parity2-parity5 / maxiter=100;

ods output Modelfit=ssdx2c(where = (criterion = "Deviance"));

output out=dx2cc xbeta=dxbc stdreschi=dx2c;run;

data dx2c;

merge dx2cc ssnx2c;

by h;

vardx2c=sqrt(valuedf);

drop criterion df value valuedf;

run;

ods select all;

proc plot data=dx2a;

title "Residual Plot A";

plot dx2a*dxba ; proc plot data=dx2c;

title "Residual Plot C";

plot dx2c*dxbc ; run;quit;run;

*NORMAL MODELS ACCOUNTING FOR HETEROSCEDASTICITY;

*METHOD IMPLEMENTED BY CERDA ET AL AS CITED IN MAIN TEXT;

ods select none;

proc reg data=a;by h;

model x2a = mage page magepage parity2 parity3 parity4 parity5;

output out=resid r=rr;

data resid;

set resid;

r2 = rr**2;

proc genmod data=resid;by h;

model r2 = mage pind / dist=normal link=log noint;

output out=x2avar p=x2avar;

proc reg data=a;by h;

model logx2c = mage page magepage parity2 parity3 parity4 parity5;

output out=resid r=rr;

data resid;

set resid;

r2 = rr**2;

proc genmod data=resid;by h;

model r2 = mage pind / dist=normal link=log noint;

output out=x2cvar p=x2cvar;*variance;

run;quit;run;

*METHOD IMPLEMENTED BY HERNAN AND ROBINS CAUSAL INFERENCE BOOK, CHAPTER 12 SECTION 4 (TRUNCATED NORMAL WITH HETEROSCEDASTIC VARIANCE);

proc qlim data=a ; by h;

model x2a = / truncated(lb=5 ub=30);

hetero x2a ~ / link = exp noconst;

nloptions tech=newrap maxiter=1000 maxfunc = 2000;

output out = numx2a_th predicted errstd ;

data numx2a_th;set numx2a_th;rename p_x2a=Pn_x2a errstd_x2a=Errstdn_x2a;run;

proc qlim data=a ; by h;

model x2a = mage page magepage parity2 parity3 parity4 parity5 / truncated(lb=5 ub=30);

hetero x2a ~ mage pind / link = exp noconst;

nloptions tech=newrap maxiter=1000 maxfunc = 2000;

output out = denx2a_th predicted errstd ;

data denx2a_th;set denx2a_th;rename p_x2a=Pd_x2a errstd_x2a=Errstdd_x2a;

proc qlim data=a ; by h;

model logx2c = / truncated(lb=-7.5 ub=3.5);

hetero logx2c ~ / link = exp noconst;

nloptions tech=newrap maxiter=1000 maxfunc = 2000;

output out = numx2c_th predicted errstd ;

data numx2c_th;set numx2c_th;rename p_logx2c=Pn_x2c errstd_logx2c=Errstdn_x2c;run;

proc qlim data=a ; by h;

model logx2c = mage page magepage parity2 parity3 parity4 parity5 / truncated(lb=-7.5 ub=3.5);

hetero logx2c ~ mage pind / link = exp noconst;

nloptions tech=newrap maxiter=1000 maxfunc = 2000;

output out = denx2c_th predicted errstd ;

data denx2c_th;set denx2c_th;rename p_logx2c=Pd_x2c errstd_logx2c=Errstdd_x2c;

run;quit;run;

*USER WRITTEN CODE FOR HETERO NORMAL;

proc nlmixed data=a tech=NRRIDG maxiter=1000 maxfunc=2000;

title "Numerator Model";

by h;

parms b0=0 sd0=1;

bounds sd0 > 0;

*MEAN;

mu = b0;

*CONSTANT VARIANCE;

sigma_sq = sd0;

sigma = sqrt(sigma_sq);

model x2a ~ normal(mu,sigma_sq);

predict mu out=bbnx2amu;

predict sigma out=bbnx2asig;

proc nlmixed data=a tech=NRRIDG maxiter=1000 maxfunc=2000;

title "Denominator Model";

by h;

parms b0=0 b1=0 b2=0 b3=0 b4=0 b5=0 b6=0 b7=0 sigma_sq=1 sd1=1 sd2=1;

bounds sigma_sq > 0, sd1>0, sd2>0;

*MEAN;

mu = b0 + b1*mage + b2*page + b3*mage*page + b4*parity2 + b5*parity3 + b6*parity4 + b7*parity5;

*HETEROSCEDASTIC VARIANCE FUNCTION;

*See, e.g., Davidian and Carroll (1987) JASA Col 82 Issue 400 for justification of equation;

sigma_i_sq = log(sigma_sq) + sd1*mage + sd2*pind;

sigma_i = sqrt(sigma_i_sq);

model x2a ~ normal(mu,sigma_i_sq);

predict mu out=bbdx2amu;

predict sigma_i out=bbdx2asig;

run;quit;run;

data bbnx2amu;set bbnx2amu;rename pred=nx2amu;run;

data bbnx2asig;set bbnx2asig;rename pred=nx2asig;run;

data bbdx2amu;set bbdx2amu;rename pred=dx2amu;run;

data bbdx2asig;set bbdx2asig;rename pred=dx2asig;run;

proc nlmixed data=a tech=nrridg maxiter=1000 maxfunc=2000;

by h;

parms b0=0 g0=0 sd0=1;

bounds sd0 > 0;

*CONTINUOUS MEAN PORTION;

mu = b0;

*ZERO INFLATED MEAN PORTION;

pi = 1/(1+exp(-(g0)));

*CONSTANT VARIANCE;

sigma_sq = sd0;

sigma = sqrt(sigma_sq);

ll = (x2c=0)*(log(pi)) + (x2c>0)*((log(1-pi))+log(pdf('normal',logx2c,mu,sigma)));

model logx2c ~ general(ll);

predict mu out=bbnx2cmu;

predict pi out=bbnx2cpi;

predict sigma out=bbnx2csig;run;

proc nlmixed data=a tech=nrridg maxiter=1000 maxfunc=2000;

by h;

parms b0=0 b1=0 b2=0 b3=0 b4=0 b5=0 b6=0 b7=0 g0=0 g1=0 g2=0

sigma_sq=1 sd1=0 sd2=0;

bounds sigma_sq > 0;

*CONTINUOUS MEAN PORTION;

mu = b0 + b1*mage + b2*page + b3*mage*page + b4*parity2 + b5*parity3 + b6*parity4 + b7*parity5;

*ZERO INFLATED MEAN PORTION;

pi = 1/(1+exp(-(g0 + g1*mage + g2*pind)));

*HETEROSCEDASTIC VARIANCE;

log_sigma_i_sq = log(sigma_sq) + sd1*mage + sd2*pind;

sigma_i = sqrt(exp(log_sigma_i_sq));

ll = (x2c=0)*log(pi) + (x2c>0)*((log(1-pi))+log(pdf('normal',logx2c,mu,sigma_i)));

model logx2c ~ general(ll);

predict mu out=bbdx2cmu;

predict pi out=bbdx2cpi;

predict sigma_i out=bbdx2csig;

run;quit;run;

data bbnx2cmu;set bbnx2cmu;rename pred=nx2cmu;keep h id pred;run;

data bbnx2cpi;set bbnx2cpi;rename pred=nx2cpi;keep h id pred;run;

data bbnx2csig;set bbnx2csig;rename pred=nx2csig;keep h id pred;run;

data bbdx2cmu;set bbdx2cmu;rename pred=dx2cmu;keep h id pred;run;

data bbdx2cpi;set bbdx2cpi;rename pred=dx2cpi;keep h id pred;run;

data bbdx2csig;set bbdx2csig;rename pred=dx2csig;keep h id pred;run;

*GAMMA MODEL;

proc genmod data=a ; by h;

title "Continuous Exposure Model: Numerator";

model x2a = / link=log dist=gamma maxiter=100;

ods output ParameterEstimates=ngx2aa1(where=(Parameter="Scale"));

output out=ngx2aa2 p=ngx2a;run;

data ngx2a;

merge ngx2aa1 ngx2aa2;

by h;

ngx2ascale = estimate;

drop parameter df stderr lowerwaldcl upperwaldcl chisq probchisq;

proc genmod data=a ; where x2c > 0;

by h;

title "Continuous Exposure Model: Numerator Continuous Portion";

model x2c = / link=log dist=gamma maxiter=100;

ods output ParameterEstimates=ngx2cc1(where=(Parameter="Scale"));

output out=ngx2cc2 p=ngx2c;

data ngx2c;

merge ngx2cc1 ngx2cc2;

by h;

ngx2cscale = estimate;

drop parameter df stderr lowerwaldcl upperwaldcl chisq probchisq;

proc genmod data=a ; by h;

title "Continuous Exposure Model: Denominator";

model x2a = mage page mage*page parity2-parity5 / link=log dist=gamma maxiter=100;

ods output ParameterEstimates=dgx2aa1(where=(Parameter="Scale"));

output out=dgx2aa2 p=dgx2a;

data dgx2a;

merge dgx2aa1 dgx2aa2;

by h;

dgx2ascale = estimate;

drop parameter df stderr lowerwaldcl upperwaldcl chisq probchisq;

proc genmod data=a ; where x2c > 0;

by h;

title "Continuous Exposure Model: Denominator Continuous Portion";

model x2c = mage page mage*page parity2-parity5 / link=log dist=gamma maxiter=100;

ods output ParameterEstimates=dgx2cc1(where=(Parameter="Scale"));

output out=dgx2cc2 p=dgx2c;run;

data dgx2c;

merge dgx2cc1 dgx2cc2;

by h;

dgx2cscale = estimate;

drop parameter df stderr lowerwaldcl upperwaldcl chisq probchisq;

run;quit;run;

*CHECK IF CV IS CONSTANT ACROSS MU (GAMMA ASSUMPTION);

data dx2a;

set dx2a;

rr = dx2a**2;

proc genmod data=dx2a;

model rr = dxba / link=log dist=normal;

output out=cv p=sigma;

run;quit;run;

data cv;

set cv;

sd = sigma**(1/2);

cv = sd/dxba;

ods select all;

proc plot data=cv;

plot cv*dxba;

run;quit;run;

ods select none;

proc reg data=cv;

model cv = dxba;

run;quit;run;

proc genmod data=a;

where x2c>0;

model x2c = mage page mage*page parity2-parity5 / maxiter=100;

output out=cv xbeta=dxbc stdreschi=dx2c;run;

data cv;

set cv;

rr = dx2c**2;

proc genmod data=cv;

where x2c>0;

model rr = dxbc / link=log dist=normal;

output out=cv p=sigma;

run;quit;run;

data cv;

set cv;

sd = sigma**(1/2);

cv = sd/(dxbc);

run;

ods select all;

proc plot data=cv;

plot cv*dxbc;

run;quit;run;

proc reg data=cv;

model cv = dxbc;

run;quit;run;

*QUANTILE BINNING APPROACH;

*RANK CONTINUOUS EXPOSURES INTO CATEGORIES;

proc rank data=a out=cc(keep= h id x2a x2c x2arank10 x2crank10) ties=low groups=10;

by h;var x2a x2c;ranks x2arank10 x2crank10;run;

proc rank data=cc out=cc ties=low groups=15;

by h;var x2a x2c;ranks x2arank15 x2crank15;run;

proc rank data=cc out=cc ties=low groups=20;

by h;var x2a x2c;ranks x2arank20 x2crank20;run;

proc sort data=a;by h id;

proc sort data=cc;by h id;

data bb;merge a cc;by h id;

data bb;set bb;

array catrank(6) x2arank10 x2crank10 x2arank15 x2crank15 x2arank20 x2crank20;

do jj = 1 to 6;

catrank(jj) = catrank(jj)+1;

end;

run;

*FIT CUMULATIVE LOGISTIC MODELS TO RANKED EXPOSURE VALUES;

proc logistic data=bb desc noprint;

by h;

title "Categorical Exposure Model: Denominator 10";

model x2arank10 = mage page mage*page parity2-parity5 / maxiter=100;

output out=aa1 predprobs=i;

run;quit;run;

data dcatx2a10;

set aa1;

array ii(*) ip_1-ip_10;

do kappa=1 to 10;

if x2arank10 = kappa then dencata10=ii(kappa);

end;

drop kappa ip_1-ip_10;

run;

proc logistic data=bb desc noprint;

by h;

title "Categorical Exposure Model: Denominator 10";

model x2crank10 = mage page mage*page parity2-parity5 / maxiter=100;

output out=aa3 predprobs=i;

run;quit;run;

data dcatx2c10;

set aa3;

array ii(*) ip_1-ip_10;

do kappa=1 to 10;

if x2crank10 = kappa then dencatc10=ii(kappa);

end;

drop kappa ip_1-ip_10;

run;

proc logistic data=bb desc noprint;

by h;

title "Categorical Exposure Model: Denominator 15";

model x2arank15 = mage page mage*page parity2-parity5 / maxiter=100;

output out=aa1 predprobs=i;

run;quit;run;

data dcatx2a15;

set aa1;

array ii(*) ip_1-ip_15;

do kappa=1 to 15;

if x2arank15 = kappa then dencata15=ii(kappa);

end;

drop kappa ip_1-ip_15;

run;

proc logistic data=bb desc noprint;

by h;

title "Categorical Exposure Model: Denominator 15";

model x2crank15 = mage page mage*page parity2-parity5 / maxiter=100;

output out=aa3 predprobs=i;

run;quit;run;

data dcatx2c15;

set aa3;

array ii(*) ip_1-ip_15;

do kappa=1 to 15;

if x2crank15 = kappa then dencatc15=ii(kappa);

end;

drop kappa ip_1-ip_15;

run;

proc logistic data=bb desc noprint;

by h;

title "Categorical Exposure Model: Denominator 20";

model x2arank20 = mage page mage*page parity2-parity5 / maxiter=100;

output out=aa1 predprobs=i;

run;quit;run;

data dcatx2a20;

set aa1;

array ii(*) ip_1-ip_20;

do kappa=1 to 20;

if x2arank20 = kappa then dencata20=ii(kappa);

end;

drop kappa ip_1-ip_20;

run;

proc logistic data=bb desc noprint;

by h;

title "Categorical Exposure Model: Denominator 20";

model x2crank20 = mage page mage*page parity2-parity5 / maxiter=100;

output out=aa3 predprobs=i;

run;quit;run;

data dcatx2c20;

set aa3;

array ii(*) ip_1-ip_20;

do kappa=1 to 20;

if x2crank20 = kappa then dencatc20=ii(kappa);

end;

drop kappa ip_1-ip_20;

run;

*CREATE STABILIZED IPWs;

data b;

merge a nx2cbin dx2cbin nx2a nx2c dx2a dx2c ngx2a ngx2c dgx2a dgx2c x2avar x2cvar numx2a_th denx2a_th numx2c_th denx2c_th dcatx2a10 dcatx2a15 dcatx2a20 dcatx2c10 dcatx2c15 dcatx2c20

bbnx2amu bbnx2asig bbdx2amu bbdx2asig

bbnx2cmu bbnx2cpi bbnx2csig bbdx2cmu bbdx2cpi bbdx2csig;

by h id;

*NORMAL;

*x2a;

numx2a=pdf("normal",x2a,nxba,varnx2a);

denx2a=pdf("normal",x2a,dxba,vardx2a);

*x2c;

*ZERO INFLATED PORTION;

if x2cind = 0 then do;

numx2c=nx2cbin + (1-nx2cbin)*pdf("normal",0,nxbc,varnx2c);

denx2c=dx2cbin + (1-dx2cbin)*pdf("normal",0,dxbc,vardx2c);

end;

*CONTINUOUS PORTION;

if x2cind = 1 then do;

numx2c=(1-nx2cbin)*pdf("normal",logx2c,nxbc,varnx2c);

denx2c=(1-dx2cbin)*pdf("normal",logx2c,dxbc,vardx2c);

end;

*GAMMA;

*x2a;

*DETERMINE GAMMA SHAPE PARAMETER (LAMBDA);

ngx2alambda = ngx2a/ngx2ascale;

dgx2alambda = dgx2a/dgx2ascale;

numgx2a = pdf("gamma",x2a,ngx2ascale,ngx2alambda);

dengx2a = pdf("gamma",x2a,dgx2ascale,dgx2alambda);

*x2c;

*DETERMINE GAMMA SHAPE PARAMETER (LAMBDA);

*ZERO INFLATED PORTION;

if x2cind = 0 then do;

numgx2c = nx2cbin;*b/c gamma pdf defined as zero when x2c = 0, no continuous portion;

dengx2c = dx2cbin;

end;

*CONTINUOUS PORTION;

if x2cind = 1 then do;

ngx2clambda = ngx2c/ngx2cscale;

dgx2clambda = dgx2c/dgx2cscale;

numgx2c = (1-nx2cbin)*pdf("gamma",x2c,ngx2cscale,ngx2clambda);

dengx2c = (1-dx2cbin)*pdf("gamma",x2c,dgx2cscale,dgx2clambda);

end;

*NORMAL HETEROSCEDASTIC: USING CERDA ET AL'S METHOD;

*x2a;

numx2ah=pdf("normal",x2a,nxba,varnx2a);

denx2ah=pdf("normal",x2a,dxba,sqrt(x2avar));

*x2c;

*ZERO INFLATED PORTION;

if x2cind = 0 then do;

numx2ch=nx2cbin;* + (1-nx2cbin)*pdf("normal",0,nxbc,varnx2c);

denx2ch=dx2cbin;* + (1-dx2cbin)*pdf("normal",0,dxbc,x2cvar);

end;

*CONTINUOUS PORTION;

if x2cind = 1 then do;

numx2ch=(1-nx2cbin)*pdf("normal",logx2c,nxbc,varnx2c);

denx2ch=(1-dx2cbin)*pdf("normal",logx2c,dxbc,sqrt(x2cvar));

end;

*NORMAL HETEROSCEDASTIC: USING HERNAN & ROBINS' METHOD;

*x2a;

nx2a_th = pdf("normal",x2a,Pn_x2a,Errstdn_x2a);

ubnx2a_th = cdf("normal",30, Pn_x2a, Errstdn_x2a);

lbnx2a_th = cdf("normal",5, Pn_x2a, Errstdn_x2a) ; numx2a_th = nx2a_th / (ubnx2a_th - lbnx2a_th);

dx2a_th = pdf("normal",x2a,Pd_x2a,Errstdd_x2a);

ubdx2a_th = cdf("normal",30, Pd_x2a, Errstdd_x2a);

lbdx2a_th = cdf("normal",5, Pd_x2a, Errstdd_x2a) ; denx2a_th = dx2a_th / (ubdx2a_th - lbdx2a_th);

*x2c;

*ZERO INFLATED PORTION;

if x2cind = 0 then do;

numx2c_th=nx2cbin + (1-

nx2cbin)*(pdf("normal",logx2c,pn_x2c,errstdn_x2c)/(cdf("normal",3.5,pd_x2c,errstdd_x2c)- cdf("normal",-7.5,pn_x2c,errstdn_x2c)));

denx2c_th=dx2cbin + (1-

dx2cbin)*(pdf("normal",logx2c,pd_x2c,errstdd_x2c)/(cdf("normal",3.5,pd_x2c,errstdd_x2c)- cdf("normal",-7.5,pd_x2c,errstdd_x2c)));

end;

*CONTINUOUS PORTION;

if x2cind = 1 then do;

numx2c_th=(1-

nx2cbin)*(pdf("normal",logx2c,pn_x2c,errstdn_x2c)/(cdf("normal",3.5,pd_x2c,errstdd_x2c)- cdf("normal",-7.5,pn_x2c,errstdn_x2c)));

denx2c_th=(1-

dx2cbin)*(pdf("normal",logx2c,pd_x2c,errstdd_x2c)/(cdf("normal",3.5,pd_x2c,errstdd_x2c)- cdf("normal",-7.5,pd_x2c,errstdd_x2c)));

end;

*NORMAL HETEROSCEDASTIC: USER WRITTEN;

*x2a;

numx2ahu=pdf("normal",x2a,nx2amu,nx2asig);

denx2ahu=pdf("normal",x2a,dx2amu,dx2asig);

*x2c;

*ZERO PORTION;

if x2c = 0 then do;

numx2chu = nx2cpi;

denx2chu = dx2cpi;

end;

*CONTINUOUS PORTION;

if x2c > 0 then do;

numx2chu = (1-nx2cpi)*pdf("normal",logx2c,nx2cmu,nx2csig);

denx2chu = (1-dx2cpi)*pdf("normal",logx2c,dx2cmu,dx2csig);

end;

*T DISTRIBUTED;

numt1x2a = pdf("T",nx2a,1);

numt1x2c = pdf("T",nx2c,1);

dent1x2a = pdf("T",dx2a,1);

dent1x2c = pdf("T",dx2c,1);

numt3x2a = pdf("T",nx2a,3);

numt3x2c = pdf("T",nx2c,3);

dent3x2a = pdf("T",dx2a,3);

dent3x2c = pdf("T",dx2c,3);

numt5x2a = pdf("T",nx2a,5);

numt5x2c = pdf("T",nx2c,5);

dent5x2a = pdf("T",dx2a,5);

dent5x2c = pdf("T",dx2c,5);

*STABILIZED WEIGHTS;

*NORMAL;

swx2a=numx2a/denx2a;

swx2c=numx2c/denx2c;

*HETEROSCEDASTIC NORMAL;

swx2ah=numx2ah/denx2ah;

swx2ch=numx2ch/denx2ch;

*TRUNCATED HETEROSCEDASTIC NORMAL;

swx2a_th = numx2a_th/denx2a_th;

swx2c_th = numx2c_th/denx2c_th;

*USER WRITTEN HETERO NORMAL;

swx2ahu = numx2ahu/denx2ahu;

swx2chu = numx2chu/denx2chu;

*GAMMA;

swgx2a=numgx2a/dengx2a;

swgx2c=numgx2c/dengx2c;

*T;

swt1x2a=numt1x2a/dent1x2a;

swt1x2c=numt1x2c/dent1x2c;

swt3x2a=numt3x2a/dent3x2a;

swt3x2c=numt3x2c/dent3x2c;

swt5x2a=numt5x2a/dent5x2a;

swt5x2c=numt5x2c/dent5x2c;

*QUANTILE BINNING APPROACH;

*BECAUSE QUANTILES ARE USED, NUMERATOR PROBABILITIES ARE SIMPLY 1/#QUANTILES;

swcatx2a10 = (1/10)/dencata10;

swcatx2a15 = (1/15)/dencata15;

swcatx2a20 = (1/20)/dencata20;

swcatx2c10 = (1/10)/dencatc10;

swcatx2c15 = (1/15)/dencatc15;

swcatx2c20 = (1/20)/dencatc20;

merg=1;

run;

*FIND TRUNCATION VALUES;

proc univariate data=b noprint;by h;var swx2a swx2c swx2ah swx2ch swx2ahu swx2chu swx2a_th

swx2c_th swgx2a swgx2c;

output out=ptile p1 =swx2a1 swx2c1 swx2ah1 swx2ch1 swx2ahu1 swx2chu1 swx2a_th1 swx2c_th1 swgx2a1 swgx2c1

p99=swx2a99 swx2c99 swx2ah99 swx2ch99 swx2ahu99 swx2chu99 swx2a_th99 swx2c_th99 swgx2a99 swgx2c99

p5 =swx2a5 swx2c5 swx2ah5 swx2ch5 swx2ahu5 swx2chu5 swx2a_th5 swx2c_th5 swgx2a5 swgx2c5

p95=swx2a95 swx2c95 swx2ah95 swx2ch95 swx2ahu95 swx2chu95 swx2a_th95 swx2c_th95 swgx2a95 swgx2c95

p10=swx2a10 swx2c10 swx2ah10 swx2ch10 swx2ahu10 swx2chu10 swx2a_th10 swx2c_th10 swgx2a10 swgx2c10

p90=swx2a90 swx2c90 swx2ah90 swx2ch90 swx2ahu90 swx2chu90 swx2a_th90 swx2c_th90 swgx2a90 swgx2c90;

run;

data ptile;set ptile;merg=1;

*TRUNCATE STABILIZED WEIGHTS FROM NORMAL AND GAMMA MODELS;

data cc;merge b ptile;by h merg;

*NORMAL X2A AND X2C;

swx2a199 = swx2a;

if swx2a < swx2a1 then swx2a199 = swx2a1;

if swx2a > swx2a99 then swx2a199 = swx2a99;

swx2c199 = swx2c;

if swx2c < swx2c1 then swx2c199 = swx2c1;

if swx2c > swx2c99 then swx2c199 = swx2c99;

swx2a595 = swx2a;

if swx2a < swx2a5 then swx2a595 = swx2a5;

if swx2a > swx2a95 then swx2a595 = swx2a95;

swx2c595 = swx2c;

if swx2c < swx2c5 then swx2c595 = swx2c5;

if swx2c > swx2c95 then swx2c595 = swx2c95;

swx2a1090 = swx2a;

if swx2a < swx2a10 then swx2a1090 = swx2a10;

if swx2a > swx2a90 then swx2a1090 = swx2a90;

swx2c1090 = swx2c;

if swx2c < swx2c10 then swx2c1090 = swx2c10;

if swx2c > swx2c90 then swx2c1090 = swx2c90;

*NORMAL HETEROSCEDASTIC X2A AND X2C;

swx2ah199 = swx2ah;

if swx2ah < swx2ah1 then swx2ah199 = swx2ah1;

if swx2ah > swx2ah99 then swx2ah199 = swx2ah99;

swx2ch199 = swx2ch;

if swx2ch < swx2ch1 then swx2ch199 = swx2ch1;

if swx2ch > swx2ch99 then swx2ch199 = swx2ch99;

swx2ah595 = swx2ah;

if swx2ah < swx2ah5 then swx2ah595 = swx2ah5;

if swx2ah > swx2ah95 then swx2ah595 = swx2ah95;

swx2ch595 = swx2ch;

if swx2ch < swx2ch5 then swx2ch595 = swx2ch5;

if swx2ch > swx2ch95 then swx2ch595 = swx2ch95;

swx2ah1090 = swx2ah;

if swx2ah < swx2ah10 then swx2ah1090 = swx2ah10;

if swx2ah > swx2ah90 then swx2ah1090 = swx2ah90;

swx2ch1090 = swx2ch;

if swx2ch < swx2ch10 then swx2ch1090 = swx2ch10;

if swx2ch > swx2ch90 then swx2ch1090 = swx2ch90;

*TRUNCATED NORMAL HETEROSCEDASTIC X2A AND X2C;

swx2a_th199 = swx2a_th;

if swx2a_th < swx2a_th1 then swx2a_th199 = swx2a_th1;

if swx2a_th > swx2a_th99 then swx2a_th199 = swx2a_th99;

swx2c_th199 = swx2c_th;

if swx2c_th < swx2c_th1 then swx2c_th199 = swx2c_th1;

if swx2c_th > swx2c_th99 then swx2c_th199 = swx2c_th99;

swx2a_th595 = swx2a_th;

if swx2a_th < swx2a_th5 then swx2a_th595 = swx2a_th5;

if swx2a_th > swx2a_th95 then swx2a_th595 = swx2a_th95;

swx2c_th595 = swx2c_th;

if swx2c_th < swx2c_th5 then swx2c_th595 = swx2c_th5;

if swx2c_th > swx2c_th95 then swx2c_th595 = swx2c_th95;

swx2a_th1090 = swx2a_th;

if swx2a_th < swx2a_th10 then swx2a_th1090 = swx2a_th10;

if swx2a_th > swx2a_th90 then swx2a_th1090 = swx2a_th90;

swx2c_th1090 = swx2c_th;

if swx2c_th < swx2c_th10 then swx2c_th1090 = swx2c_th10;

if swx2c_th > swx2c_th90 then swx2c_th1090 = swx2c_th90;

*USER WRITTEN NORMAL HETEROSCEDASTIC X2A AND X2C;

swx2ahu199 = swx2ahu;

if swx2ahu < swx2ahu1 thuen swx2ahu199 = swx2ahu1;

if swx2ahu > swx2ahu99 thuen swx2ahu199 = swx2ahu99;

swx2chu199 = swx2chu;

if swx2chu < swx2chu1 then swx2chu199 = swx2chu1;

if swx2chu > swx2chu99 then swx2chu199 = swx2chu99;

swx2ahu595 = swx2ahu;

if swx2ahu < swx2ahu5 then swx2ahu595 = swx2ahu5;

if swx2ahu > swx2ahu95 then swx2ahu595 = swx2ahu95;

swx2chu595 = swx2chu;

if swx2chu < swx2chu5 then swx2chu595 = swx2chu5;

if swx2chu > swx2chu95 then swx2chu595 = swx2chu95;

swx2ahu1090 = swx2ahu;

if swx2ahu < swx2ahu10 then swx2ahu1090 = swx2ahu10;

if swx2ahu > swx2ahu90 then swx2ahu1090 = swx2ahu90;

swx2chu1090 = swx2chu;

if swx2chu < swx2chu10 then swx2chu1090 = swx2chu10;

if swx2chu > swx2chu90 then swx2chu1090 = swx2chu90;

*GAMMA X2A AND X2C;

swgx2a199 = swgx2a;

if swgx2a < swgx2a1 then swgx2a199 = swgx2a1;

if swgx2a > swgx2a99 then swgx2a199 = swgx2a99;

swgx2a595 = swgx2a;

if swgx2a < swgx2a5 then swgx2a595 = swgx2a5;

if swgx2a > swgx2a95 then swgx2a595 = swgx2a95;

swgx2a1090 = swgx2a;

if swgx2a < swgx2a10 then swgx2a1090 = swgx2a10;

if swgx2a > swgx2a90 then swgx2a1090 = swgx2a90;

swgx2c199 = swgx2c;

if swgx2c < swgx2c1 then swgx2c199 = swgx2c1;

if swgx2c > swgx2c99 then swgx2c199 = swgx2c99;

swgx2c595 = swgx2c;

if swgx2c < swgx2c5 then swgx2c595 = swgx2c5;

if swgx2c > swgx2c95 then swgx2c595 = swgx2c95;

swgx2c1090 = swgx2c;

if swgx2c < swgx2c10 then swgx2c1090 = swgx2c10;

if swgx2c > swgx2c90 then swgx2c1090 = swgx2c90;

run;

data c;set cc;keep swx2a swx2c swx2ah swx2ch swx2ahu swx2chu swx2a_th swx2c_th

swx2a199 swx2a595 swx2a1090 swx2c199 swx2c595 swx2c1090 swx2ah199 swx2ah595 swx2ah1090 swx2ch199 swx2ch595 swx2ch1090 swx2ahu199 swx2ahu595 swx2ahu1090 swx2chu199 swx2chu595 swx2chu1090 swx2a_th199 swx2a_th595 swx2a_th1090 swx2c_th199 swx2c_th595 swx2c_th1090 swgx2a swgx2c

swgx2a199 swgx2a595 swgx2a1090 swgx2c199 swgx2c595 swgx2c1090 swt1x2a swt1x2c

swt3x2a swt3x2c swt5x2a swt5x2c swcatx2a10 swcatx2c10 swcatx2a15 swcatx2c15 swcatx2a20 swcatx2c20 x2a y2a x2c y2c h id ;run;

ods select all;

proc means data=c min max mean nmiss maxdec=2;

title "Distribution of Weights";

var

swx2a swx2c swx2ah swx2ch swx2ahu swx2chu swx2a_th swx2c_th

swx2a199 swx2a595 swx2a1090 swx2c199 swx2c595 swx2c1090 swx2ah199 swx2ah595 swx2ah1090 swx2ch199 swx2ch595 swx2ch1090 swx2ahu199 swx2ahu595 swx2ahu1090 swx2chu199 swx2chu595 swx2chu1090 swx2a_th199 swx2a_th595 swx2a_th1090

swx2c_th199 swx2c_th595 swx2c_th1090 swgx2a swgx2c

swgx2a199 swgx2a595 swgx2a1090 swgx2c199 swgx2c595 swgx2c1090 swt1x2a swt1x2c

swt3x2a swt3x2c swt5x2a swt5x2c swcatx2a10 swcatx2c10 swcatx2a15 swcatx2c15 swcatx2a20 swcatx2c20 ; run;quit;run;

*FINAL MODELS;

ods select none;

title "x2a";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal";

model y2a = x2a / dist=bin maxiter=100;

weight swx2a;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta1(where=(parm = "x2a"));run;

data beta1;set beta1;length type $ 10;type = "N";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal (1,99)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2a199;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta2(where=(parm = "x2a"));run;

data beta2;set beta2;length type $ 10;type = "N(1,99)";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal (5,95)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2a595;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta3(where=(parm = "x2a"));run;

data beta3;set beta3;length type $ 10;type = "N(5,95)";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal (10,90)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2a1090;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta4(where=(parm = "x2a"));run;

data beta4;set beta4;length type $ 10;type = "N(10,90)";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal Heteroscedastic (Cerda)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2ah;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta4a(where=(parm = "x2a"));run;

data beta4a;set beta4a;length type $ 10;type = "NHc";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal Heteroscedastic (Cerda)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2ah199;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta4a1(where=(parm = "x2a"));run;

data beta4a1;set beta4a1;length type $ 10;type = "NHc(1,99)";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal Heteroscedastic (Cerda)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2ah595;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta4a2(where=(parm = "x2a"));run;

data beta4a2;set beta4a2;length type $ 10;type = "NHc(5,95)";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal Heteroscedastic (Cerda)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2ah1090;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta4a3(where=(parm = "x2a"));run;

data beta4a3;set beta4a3;length type $ 10;type = "NHc(10,90)";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal Heteroscedastic (Hernan)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2a_th;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta4a4(where=(parm = "x2a"));run;

data beta4a4;set beta4a4;length type $ 10;type = "NHh";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal Heteroscedastic (Hernan)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2a_th199;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta4a5(where=(parm = "x2a"));run;

data beta4a5;set beta4a5;length type $ 10;type = "NHh(1,99)";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal Heteroscedastic (Hernan)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2a_th595;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta4a6(where=(parm = "x2a"));run;

data beta4a6;set beta4a6;length type $ 10;type = "NHh(5,95)";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal Heteroscedastic (Hernan)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2a_th1090;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta4a7(where=(parm = "x2a"));run;

data beta4a7;set beta4a7;length type $ 10;type = "NHh(10,90)";

proc genmod data=c desc;

by h;class id;

title2 "Standard Normal Heteroscedastic (Naimi)";

model y2a = x2a / dist=bin maxiter=100;

weight swx2ahu;

repeated subject=id / type=ind;

ods output GEEEmpPEst=beta4a8(where=(parm = "x2a"));run;

data beta4a8;set beta4a8;length type $ 10;type = "NHu";

proc genmod data=c desc;

by h;class id;